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atomic formula

Let Σnormal-Σ\Sigma be a signature and T⁢(Σ)Tnormal-ΣT(\Sigma) the set of terms over Σnormal-Σ\Sigma. The set SSS of symbols for T⁢(Σ)Tnormal-ΣT(\Sigma) is the disjoint union of Σnormal-Σ\Sigma and VVV, a countably infinite set whose elements are called variables. Now, adjoin SSS the set {=,(,)}fragmentsnormal-{normal-,fragmentsnormal-(normal-,normal-)normal-}\{=,(,)\}, assumed to be disjoint from SSS. An atomic formulaφφ\varphi over Σnormal-Σ\Sigma is any one of the following:

1.

either (t1=t2)subscriptt1subscriptt2(t_{1}=t_{2}), where t1subscriptt1t_{1} and t2subscriptt2t_{2} are terms in T⁢(Σ)Tnormal-ΣT(\Sigma),

2.

or (R⁢(t1,…,tn))Rsubscriptt1normal-…subscripttn(R(t_{1},...,t_{n})), where R∈ΣRnormal-ΣR\in\Sigma is an nnn-aryrelation symbol, and ti∈T⁢(Σ)subscripttiTnormal-Σt_{i}\in T(\Sigma).

Remarks.

1.

Using atomic formulas, one can inductively build formulas using the logical connectives∨\vee, ¬\neg, ∃\exists, etc… In this sense, atomic formulas are formulas that can not be broken down into simpler formulas; they are the building blocks of formulas.

2.

A literal is a formula that is either atomic or of the form ¬⁢φφ\neg\varphi where φφ\varphi is atomic. If a literal is atomic, it is called a positive literal. Otherwise, it is a negative literal.

3.

A finitedisjunction of literals is called a clause. In other words, a clause is a formula of the form φ1∨φ2∨⋯∨φnsubscriptφ1subscriptφ2normal-⋯subscriptφn\varphi_{1}\vee\varphi_{2}\vee\cdots\vee\varphi_{n}, where each φisubscriptφi\varphi_{i} is a literal.

4.

A qunatifier-free formula is a formula that does not contain the symbols ∃\exists or ∀for-all\forall.